On the choice of time in the continuum limit of polymeric effective theories

TL;DR

This paper investigates the continuum limit in polymeric quantum theories, showing that a specific $mbda$-dependent internal time is necessary for a well-defined limit, and compares classical and quantum approaches.

Contribution

It introduces a $mbda$-dependent internal time to define the continuum limit in classical loop quantum cosmology and discusses its implications for the quantum theory.

Findings

The classical limit exists only with a $mbda$-dependent internal time.

Quantum dynamics with a $mbda$-independent parameter lacks a well-defined limit.

A consistent quantum time parameter may be constructed based on the classical analysis.

Abstract

In polymeric quantum theories, a natural question pertains to the so called continuum limit, corresponding to the limit where the `discreteness parameter' $\lambda$ approaches zero. In particular one might ask whether the limit exists and, in that case, what the limiting theory is. Here we review recent results on the classical formulation of the problem for a soluble model in loop quantum cosmology. We show that it is only through the introduction of a particular $\lambda$-dependent internal time function that the limit $\lambda\to 0$ can be well defined. We then compare this result with the existing analysis in the quantum theory, where the dynamics was cast in terms of an internal ($\lambda$-independent) parameter for which the limit does not exist. We briefly comment on the steps needed to define the corresponding time parameter in the quantum theory for which the limit was shown to exist classically.